Abstract
In this paper, we develop an abstract framework to establish ill-posedness, in the sense of Hadamard, for some nonlocal PDEs displaying unbounded unstable spectra. We apply this to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov–Dirac–Benney system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bardos, C.: About a variant of the 1d Vlasov equation, dubbed “Vlasov–Dirac–Benney” equation. Séminaire Laurent Schwartz—EDP et Appl. 15: 21 (2012–2013)
Bardos C., Besse N.: The Cauchy problem for the Vlasov–Dirac–Benney equation and related issues in fluid mechanics and semi-classical limits. Kinet. Relat. Models 6(4), 893–917 (2013)
Bardos, C., Nouri, A.: A Vlasov equation with Dirac potential used in fusion plasmas. J. Math. Phys. 53(11), 115621, 16 (2012)
Benzoni-Gavage S., Coulombel J.-F., Tzvetkov N.: Ill-posedness of nonlocal Burgers equations. Adv. Math. 227(6), 2220–2240 (2011)
Bossy M., Fontbona J., Jabin P.-E., Jabir J.-F.: Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain. Commun. Partial Differ. Equ. 38(7), 1141–1182 (2013)
Brenier, Y.: A Vlasov–Poisson type formulation of the Euler equations for perfect incompressible fluids. Rapp. de recherche INRIA (1989)
Brenier Y.: Homogeneous hydrostatic flows with convex velocity profiles. Nonlinearity 12(3), 495–512 (1999)
Brenier Y.: Convergence of the Vlasov–Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 25(3–4), 737–754 (2000)
Brenier Y.: Remarks on the derivation of the hydrostatic Euler equations. Bull. Sci. Math. 127(7), 585–595 (2003)
Brenier Y.: A homogenized model for vortex sheets. Arch. Ration. Mech. Anal. 138(4), 319–353 (1997)
Caflisch R.E.: A simplified version of the abstract Cauchy–Kowalewski theorem with weak singularities. Bull. Am. Math. Soc. (N.S.) 23(2), 495–500 (1990)
Chen, X.L., Morrison, P.J.: A sufficient condition for the ideal instability of shear flow with parallel magnetic field. Phys. Fluids B Plasma Phys. (1989–1993) 3(4): 863–865 (1991)
Desjardins B., Grenier E.: On nonlinear Rayleigh–Taylor instabilities. Acta Math. Sin. (Engl. Ser.) 22(4), 1007–1016 (2006)
Friedlander S., Vicol V.: On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. Nonlinearity 24(11), 3019–3042 (2011)
Gérard-Varet D., Nguyen T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77(1–2), 71–88 (2012)
Gérard-Varet D., Dormy E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)
Grenier E.: Defect measures of the Vlasov–Poisson system in the quasineutral regime. Commun. Partial Differ. Equ. 20(7–8), 1189–1215 (1995)
Grenier E.: On the derivation of homogeneous hydrostatic equations. M2AN Math. Model. Numer. Anal 33(5), 965–970 (1999)
Grenier, E., Guo, Y., Nguyen, T.: Spectral instability of symmetric shear flows in a two-dimensional channel. Adv. Math. 292, 52–110 (2016)
Guo Y., Nguyen T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64(10), 1416–1438 (2011)
Guo Y., Tice I.: Compressible, inviscid Rayleigh–Taylor instability. Indiana Univ. Math. J. 60(2), 677–711 (2011)
Hadamard J.: Lectures on Cauchy’s problem in linear partial differential equations. Dover Publications, New York (1953)
Han-Kwan, D., Iacobelli, M.: The quasineutral limit of the Vlasov–Poisson equation in Wasserstein metric. Commun. Math. Sci. (2014). (To appear)
Han-Kwan, D., Iacobelli, M.: Quasineutral limit for Vlasov–Poisson via Wasserstein stability estimates in higher dimension (2015). (Submitted)
Han-Kwan, D., Nguyen, T.: Nonlinear instability of Vlasov–Maxwell systems in the classical and quasineutral limits (2015). (Preprint)
Han-Kwan, D., Rousset, F.: Quasineutral limit for Vlasov–Poisson with Penrose stable data. Ann. Sci. Éc. Norm. Supér. (2015). (To appear)
Han-Kwan D.: Quasineutral limit of the Vlasov–Poisson system with massless electrons. Commun. Partial Differ. Equ. 36(8), 1385–1425 (2011)
Han-Kwan D., Hauray M.: Stability issues in the quasineutral limit of the one-dimensional Vlasov–Poisson equation. Commun. Math. Phys. 334(2), 1101–1152 (2015)
Jabin P.-E., Nouri A.: Analytic solutions to a strongly nonlinear Vlasov equation. C. R. Math. Acad. Sci. Paris 349(9–10), 541–546 (2011)
Kukavica, I., Masmoudi, N., Vicol, V., Kwong Wong, T.: On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46(6), 3865–3890 (2014)
Kukavica I., Temam R., Vicol V.C., Ziane M.: Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. J. Differ. Equ. 250(3), 1719–1746 (2011)
Lax P.D.: Nonlinear hyperbolic equations. Commun. Pure Appl. Math. 6, 231–258 (1953)
Lebeau, G.: Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2d. ESAIM Control Optim. Calc. Var. 8, 801–825 (2002). [A tribute to J. L. Lions (electronic)]
Lerner, N., Nguyen, T., Texier, B.: The onset of instability in first-order systems. arXiv:1504.04477
Lions, P.-L.: Mathematical topics in fluid mechanics. Volume 3 of Oxford Lecture Series in Mathematics and its Applications, Vol. 1. The Clarendon Press, Oxford University Press, New York, 1996. (Incompressible models, Oxford Science Publications)
Masmoudi N., Wong T.K.: On the H s theory of hydrostatic Euler equations. Arch. Ration. Mech. Anal. 204(1), 231–271 (2012)
Métivier, G.: Remarks on the well-posedness of the nonlinear Cauchy problem. Geometric Analysis of PDE and Several Complex Variables. Contemp. Math., Vol. 368. Am. Math. Soc., Providence, 337–356, 2005
Mizohata, S.: Some remarks on the Cauchy problem. J. Math. Kyoto Univ. 1, 109–127 (1961/1962)
Mouhot C., Villani C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, Vol. 44. Springer, New York, 1983
Renardy M.: Ill-posedness of the hydrostatic Euler and Navier–Stokes equations. Arch. Ration. Mech. Anal. 194(3), 877–886 (2009)
Seiichiro W.: The Lax–Mizohata theorem for nonlinear Cauchy problems. Commun. Partial Differ. Equ. 26(7–8), 1367–1384 (2001)
Zumbrun, K.: Planar stability criteria for viscous shock waves of systems with real viscosity. Hyperbolic Systems of Balance Laws. Lecture Notes in Math., Vol. 1911. Springer, Berlin, 229–326, 2007
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
This paper is dedicated to Claude Bardos on the occasion of his 75th birthday, as a token of friendship and admiration
Rights and permissions
About this article
Cite this article
Han-Kwan, D., Nguyen, T.T. Ill-Posedness of the Hydrostatic Euler and Singular Vlasov Equations. Arch Rational Mech Anal 221, 1317–1344 (2016). https://doi.org/10.1007/s00205-016-0985-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-0985-z